50 Mr. A. Mullock. On a Direct Method of Measuring [Apr. 19, 



On a Direct Method of Measuring the Coefficient of Volume 

 Elasticity of Metals." Uy A. MALLOCK, F.RS. Received 

 April 19, Read June 2, 1904. 



For most hard materials the coefficient of volume elasticity is usually 

 calculated from measurements of Young's modulus or of the coefficient 

 of rigidity, either of which, when Poisson's ratio is known, suffices for 

 iU determination. Although, however, the total alteration of volume 

 for a given pressure can he calculated from the coefficient thus obtained, 

 it is only for isotropic material that the alteration of dimensions in any 

 given direction can lie inferred from it. 



The following direct method of measuring the coefficient of volume 

 elasticity is of some interest, as it allows of the linear contraction or 

 extension in any given direction, caused in a substance by fluid pressure, 

 to be measured independently of other changes of form. 



When a long circular cylinder is acted on by internal fluid pressure, 

 if the walls are very thin compared to the diameter of the cylinder, 

 the stress in the material parallel to the axis is just half the stress at 

 right angles to the axis in the tangent plane. The conditions of strain 

 and stress in the walls of the cylinder can be expressed in terms of the 

 volume elasticity and rigidity of the material as follows : 



Consider a small cube of the material, with edges parallel to the axis 

 (X), radius (Y), and tangent of circular section (Z) of the cylinder 

 respectively. Let K be that part of the stress which produces 

 alteration of volume, and N and N yz the parts which produce shear in 

 the planes XZ and YZ. 



The total force acting parallel to Y, i.e., in the direction of the 

 radius, vanishes in comparison with the forces at right angles to it 

 when the walls of the cylinder are thin, for while the radial force is 

 at one surface of the cylinder, and equal per unit area to the fluid 

 pi-ossure (P) on the other, in the two directions at right angles to the 



Jius the stress is of the order Pr// (= 2F), where r is the radius and 

 / the thickness of the wall. 



Hence we have the following relations between K, N, and P' : 



K + N-w + Nj,, = 2F ....... n\ 



K-N X , = F ....... ...... (2) 



K - 



s 



